Prove that √3 is an irrational number.

By BYJU'S Exam Prep

Updated on: September 25th, 2023

Consider that √3 is a rational number. Then there are positive integers a and b such that √3 = a/b, where a and b are co-prime, meaning their HCF is 1.

√3 = a/b

3 = a2/b2

3b2 = a2

3 divides a2 [As 3 divides 3b2]

3 divides a …..(1)

a = 3c for some integer c

a2 = 9c2

3b2 = 9c2 [As a2 = 3b2]

b2 = 3c2

3 divides b2 [As 3 divides c2]

3 divides b …. (2)

From (1) and (2), it is clear that a and b share at least 3 as common components (2).

However, this is contradicted by the fact that a and b are co-prime, which shows that our theory is flawed.

Hence, √3 is an irrational number.

List of Irrational Numbers

  • Pi, Euler’s number, and the Golden ratio are examples of notable irrational numbers.
  • Not all square roots, cube roots, and other numbers exhibit irrationality.
  • For example, √3 is an irrational number, but √4 is a rational number.
  • Because 4 is a perfect square, such as 4 = 2 x 2 and √4 = 2, which is a rational number.
  • It should be emphasised that between any two real numbers, there are infinitely many irrational numbers.
  • For instance, between two integers, let’s say 1 and 2, there exists an unlimited amount of irrational numbers.


Prove that √3 is an irrational number.

It is proved that √3 is an irrational number. The other examples of irrational numbers are Pi, Euler’s number, and the Golden ratio.

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